Let $\mathbb Q_{>0}$ be the set of all positive rational numbers. Let $f:\mathbb Q_{>0}\to\mathbb R$ be a function satisfying the following three conditions:
(i) for all $x,y\in\mathbb Q_{>0}$, we have $f(x)f(y)\geq f(xy)$;
(ii) for all $x,y\in\mathbb Q_{>0}$, we have $f(x+y)\geq f(x)+f(y)$;
(iii) there exists a rational number $a>1$ such that $f(a)=a$.
Prove that $f(x)=x$ for all $x\in\mathbb Q_{>0}$.
Proposed by Bulgaria
Let $n \ge 3$ be an integer, and consider a circle with $n + 1$ equally spaced points marked on it. Consider all labellings of these points with the numbers $0, 1, ... , n$ such that each label is used exactly once; two such labellings are considered to be the same if one can be obtained from the other by a rotation of the circle. A labelling is called beautiful if, for any four labels $a < b < c < d$ with $a + d = b + c$, the chord joining the points labelled $a$ and $d$ does not intersect the chord joining the points labelled $b$ and $c$.
Let $M$ be the number of beautiful labelings, and let N be the number of ordered pairs $(x, y)$ of positive integers such that $x + y \le n$ and $\gcd(x, y) = 1$. Prove that $$M = N + 1.$$
Given triangle $ABC$ the point $J$ is the centre of the excircle opposite the vertex $A.$ This excircle is tangent to the side $BC$ at $M$, and to the lines $AB$ and $AC$ at $K$ and $L$, respectively. The lines $LM$ and $BJ$ meet at $F$, and the lines $KM$ and $CJ$ meet at $G.$ Let $S$ be the point of intersection of the lines $AF$ and $BC$, and let $T$ be the point of intersection of the lines $AG$ and $BC.$ Prove that $M$ is the midpoint of $ST.$
(The excircle of $ABC$ opposite the vertex $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.)
Proposed by Evangelos Psychas, Greece
Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$. Prove that
\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]
Proposed by Angelo Di Pasquale, Australia
The liar's guessing game is a game played between two players $A$ and $B$. The rules of the game depend on two positive integers $k$ and $n$ which are known to both players.
At the start of the game $A$ chooses integers $x$ and $N$ with $1 \le x \le N.$ Player $A$ keeps $x$ secret, and truthfully tells $N$ to player $B$. Player $B$ now tries to obtain information about $x$ by asking player $A$ questions as follows: each question consists of $B$ specifying an arbitrary set $S$ of positive integers (possibly one specified in some previous question), and asking $A$ whether $x$ belongs to $S$. Player $B$ may ask as many questions as he wishes. After each question, player $A$ must immediately answer it with yes or no, but is allowed to lie as many times as she wants; the only restriction is that, among any $k+1$ consecutive answers, at least one answer must be truthful.
After $B$ has asked as many questions as he wants, he must specify a set $X$ of at most $n$ positive integers. If $x$ belongs to $X$, then $B$ wins; otherwise, he loses. Prove that:
1. If $n \ge 2^k,$ then $B$ can guarantee a win.
2. For all sufficiently large $k$, there exists an integer $n \ge (1.99)^k$ such that $B$ cannot guarantee a win.
Proposed by David Arthur, Canada
Find all functions $f:\mathbb Z\rightarrow \mathbb Z$ such that, for all integers $a,b,c$ that satisfy $a+b+c=0$, the following equality holds:
\[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\]
(Here $\mathbb{Z}$ denotes the set of integers.)
Proposed by Liam Baker, South Africa
Let $ABC$ be a triangle with $\angle BCA=90^{\circ}$, and let $D$ be the foot of the altitude from $C$. Let $X$ be a point in the interior of the segment $CD$. Let $K$ be the point on the segment $AX$ such that $BK=BC$. Similarly, let $L$ be the point on the segment $BX$ such that $AL=AC$. Let $M$ be the point of intersection of $AL$ and $BK$.
Show that $MK=ML$.
Proposed by Josef Tkadlec, Czech Republic
Find all positive integers $n$ for which there exist non-negative integers $a_1, a_2, \ldots, a_n$ such that
\[
\frac{1}{2^{a_1}} + \frac{1}{2^{a_2}} + \cdots + \frac{1}{2^{a_n}} =
\frac{1}{3^{a_1}} + \frac{2}{3^{a_2}} + \cdots + \frac{n}{3^{a_n}} = 1.
\]
Proposed by Dusan Djukic, Serbia
Show that every integer $k > 1$ has a multiple which is less than $k^4$ and can be written in base 10 using at most 4 different digits.
In a sports contest, there were $m$ medals awarded on $n$ successive days ($n > 1$). On the first day, one medal and $1/7$ of the remaining $m − 1$ medals were awarded. On the second day, two medals and $1/7$ of the now remaining
medals were awarded; and so on. On the $n^{th}$ and last day, the remaining $n$ medals were awarded. How many days did the contest last, and how many medals were awarded altogether?
For what real values of $x$ is \[ \sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A \] given
a) $A=\sqrt{2}$;
b) $A=1$;
c) $A=2$,
where only non-negative real numbers are admitted for square roots?
Determine all three-digit numbers $N$ having the property that $N$ is divisible by 11, and $\dfrac{N}{11}$ is equal to the sum of the squares of the digits of $N$.
For each integer $a_0 >$ 1, define the sequence $a_0, a_1, a_2, \cdots$ by:
$$
a_{n+1} =
\left\{
\begin{array}{ll}
\sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer}\\
a_n + 3 & \text{otherwise}
\end{array}
\right.
$$
For all $n \ge 0$. Determine all values of $a_0$ for which there is a number $A$ such that $a_n = A$ for infinitely many values of $n$.
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that, for all real numbers $x$ and $y$,
$$f (f(x)f(y)) + f(x + y) = f(xy)$$
A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit’s starting point, $A_0$, and the hunter’s starting point, $B_0$, are the same. After $n-1$ rounds of the game, the rabbit is at point $A_{n-1}$ and the hunter is at point $B_{n-1}$. In the nth round of the game, three things occur in order.
(i) The rabbit moves invisibly to a point $A_n$ such that the distance between $A_{n-1}$ and $A_n$ is exactly $1$.
(ii) A tracking device reports a point $P_n$ to the hunter. The only guarantee provided by the tracking device to the hunter is that the distance between $P_n$ and $A_n$ is at most 1.
(iii) The hunter moves visibly to a point $B_n$ such that the distance between $B_{n-1}$ and $B_n$ is exactly 1.
Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after 109 rounds she can ensure that the distance between her and the rabbit is at most 100?
Language:
Let $R$ and $S$ be different points on a circle $\Omega$ such that $RS$ is not a diameter. Let $\ell$ be the tangent line to $\Omega$ at $R$. Point $T$ is such that $S$ is the midpoint of the line segment $RT$. Point $J$ is chosen on the shorter arc $RS$ of $\Omega$ so that the circumcircle $\Gamma$ of triangle $JST$ intersects $\ell$ at two distinct points. Let $A$ be the common point of $\Gamma$ and $\ell$ that is closer to $R$. Line $AJ$ meets again
at $K$. Prove that the line $KT$ is tangent to $\Gamma$.
An integer $N > 2$ is given. A collection of $N(N + 1)$ soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove $N(N-1)$ players from this row leaving a new row of $2N$ players in which the following $N$ conditions hold:
(1) no one stands between the two tallest players,
(2) no one stands between the third and fourth tallest players,
...
(N) no one stands between the two shortest players.
Show that this is always possible.
An ordered pair $(x, y)$ of integers is a primitive point if the greatest common divisor of $x$ and $y$ is 1. Given a finite set $S$ of primitive points, prove that there exist a positive integer $n$ and integers $a_0$, $a_1$, $\cdots$, an such that, for each $(x, y)$ in $S$, we have:
$$a_0x^n + a_1x^{n-1}y + a_2x^{n-2}y^2 + \cdots + a_{n-1}xy^{n-1} + a_ny^n = 1$$
Let $n$ be a positive integer. Each point $(x,y)$ in the plane, where $x$ and $y$ are non-negative integers with $x+y < n$, is colored red or blue, subject to the following condition:if a point $(x,y)$ is red, then so are all the points $(x',y')$ with $x'\le x$ and $y'\le y$ Let $A$ be the number of ways choose $n$ blue points with distinct $x$-coordinates. and let B be the number of ways to choose $n$ blue points with distinct $y$-coordinates. Prove that $A=B$
Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line.
Find all numbers $n \ge 3$ for which there exists real numbers $a_1, a_2, ..., a_{n+2}$ satisfying $a_{n+1} = a_1, a_{n+2} = a_2$ and\[a_{i}a_{i+1} + 1 = a_{i+2}\]for $i = 1, 2, ..., n.$
An anti-Pascal triangle is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. For example, the following is an anti-Pascal triangle with four rows which contains every integer from $1$ to $10$
\[4\]\[2\quad 6\]\[5\quad 7 \quad 1\]\[8\quad 3 \quad 10 \quad 9\]
Does there exist an anti-Pascal triangle with $2018$ rows which contains every integer from $1$ to $1 + 2 + 3 + \dots + 2018$?
A site is any point $(x, y)$ in the plane such that $x$ and $y$ are both positive integers less than or equal to 20. Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to $\sqrt{5}$. On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) They stop as soon as a player cannot place a stone. Find the greatest $K$ such that Amy can ensure that she places at least $K$ red stones, no matter how Ben places his blue stones.
Let $a_1, a_2, \dots$ be an infinite sequence of positive integers. Suppose that there is an integer$N > 1$ such that, for each $n \geq N$, the number $\frac{a_1}{a_2}+\frac{a_2}{a_3}+\dots +\frac{a_{n-1}}{a_n}+\frac{a_n}{a_1}$ is an integer. Prove that there is a positive integer $M$ such that $a_m = a_{m+1}$ for all $m \geq M.$
A convex quadrilateral $ABCD$ satisfies $AB\cdot CD=BC \cdot DA.$ Point $X$ lies inside $ABCD$ so that $\angle XAB = \angle XCD$ and $\angle XBC = \angle XDA.$ Prove that $\angle BXA + \angle DXC = 180^{\circ}$ .